If A is a square matrix and there exists a square matrix B such that AB=1, than it is known that BA=1. This property is proved with some properties from linear algebra. Although I've never seen it be proved just by structures of matrix multiplication, I couldn't find a counterexample of a set with structures of matrix multiplication but left inverse doesn't imply right inverse.
To be more specific, let X be a set and binary operation ⋅ is defined on X. If ⋅ is associative and X has left and right identity(which will be the same), than does A⋅B=1 for some A,B∈X implies B⋅A=1?
If not, what other properties of matrix multiplication should we add to this structure of (X,⋅) in order to get the property?
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